\newproblem{lay:1_8_34}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.8.34}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Let $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be the transformation that reflects each vector $\mathbf{x}=(x_1,x_2,x_3)$ through the plane $x_3=0$ onto
	$T(\mathbf{x})=(x_1,x_2,-x_3)$. Show that $T$ is a linear transformation.
}{
  % Solution
	We need to show that $\forall \mathbf{u},\mathbf{v}\in\mathbb{R}^3$, $\forall c\in\mathbb{R}$
		\begin{itemize}
			\item $f(\mathbf{u}+\mathbf{v})=f(\mathbf{u})+f(\mathbf{v})$ \\
						In this particular case:
						\begin{center}
							$T(\mathbf{u}+\mathbf{v})=T((u_1+v_1,u_2+v_2,u_3+v_3))=(u_1+v_1,u_2+v_2,-u_3-v_3)$
						\end{center}
						On the other hand:
						\begin{center}
							$T(\mathbf{u})+T(\mathbf{v})=(u_1,u_2,-u_3)+(v_1,v_2,-v_3)=(u_1+v_1,u_2+v_2,-u_3-v_3)$
						\end{center}
						which are obviously equal.
			\item $T(c\mathbf{u})=cT(\mathbf{u})$ \\
						In this particular case:
						\begin{center}
							$T(c\mathbf{u})=T((cu_1,cu_2,cu_3))=(cu_1,cu_2,-cu_3)=c(u_1,u_2,-u_3)=cT(\mathbf{u})$
						\end{center}
		\end{itemize}
}
\useproblem{lay:1_8_34}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
